Nonlinear wave propagation in coaxial shells filled with viscous liquid

The investigation of deformation wave behavior in elastic shells is one of the significant trends in contemporary wave dynamics. There are wave motion mathematical models of infinitely long geometrically and physically nonlinear shells with viscous incompressible liquid inside. They are based on the coupled hydroelastic problems, described by the equations of dynamics of shells and the equations of viscous incompressible fluid in the form of generalized KdV equations. Mathematical models of wave processes in infinitely long geometrically nonlinear coaxial cylindrical shells are obtained by means of the small parameter perturbation method. The problems differ from the already known ones by the consideration of viscous incompressible liquid. The system of generalized KdV equations is obtained on the basis of coupled hydroelastic problems, described by the equations of dynamics of shells and fluid equations with corresponding boundary conditions. The article deals with investigating the model describing wave phenomena in two geometrically and physically nonlinear elastic coaxial cylindrical Kirchhoff-Love type shells, containing viscous incompressible liquid both between and inside them. The difference Crank-Nicholson type schemes aimed at investigating equations systems with the consideration of liquid impact are obtained with the help of Gröbner basis construction. To generate these difference schemes, basic integral difference correlations, approximating the initial equation system, are used. The use of Gröbner basis techniques makes it possible to generate the schemes allowing one to obtain discrete preservation law analogues to initial differential equations. To do this, equivalent transformations were made. On the basis of computational algorithm the program complex allowing one to construct graphs and to obtain Cauchy problem numerical solution was developed using the exact solutions of the system of coaxial shell dynamics equations as an initial condition.